Open. No proof or disproof of the statement for an arbitrary sum-free $S\subseteq\mathbb R$ was found in the search whose scope the Current assessment records. Two special cases are not the problem. The case $S$ Sidon (the site's variant) is claimed by an argument that AlphaProof found, merged as a Lean proof into the formal-conjectures statement file on 6 January 2026, posted to the site's discussion on 7 January 2026 and adopted by the site's commentary; the first case of that argument also covers every $S$ of cardinality less than $\mathfrak c$ without the Sidon hypothesis. It is recorded as a pending partial claim on its claim page (Deepmind, 2026), which the commentary on an open problem does not make accepted. The case of $S$ with the property of Baire is argued in a thread comment of 23 January 2026, not adopted by the commentary; it is a thread post, not a dated manuscript or a Lean proof, so it has no claim page. The proof-claim tab is empty. This is a bounded negative finding, not a certificate of openness.